No, to the best of my knowledge if you work a 6 hour shift (5 in some states), you are entitled to a 30 min lunch break no matter if you work part time or full time at your store.

Beware of the dreaded 5:30 length shift. Absolutely awful

6+ hours gets a lunch at corporate locations no matter what state. Being a part-time or full-time does not affect that.

Now if you work at a licensed location like in a Target or some grocery store, that may be a different story. You will have to follow the policy of that store, not Starbucks policy. However depending on your state, you still may be legally entitled to a lunch.